#237146

order	512
nilpotency class	3
isoclinism	isoclinic class of #237115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1163 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,10)(3,11,14,16)(4,8,6,9)(7,12,15,13)(17,18,23,28)(19,29,31,41)(20,34,38,43)(21,27,22,26)(24,33,40,32)(25,36,44,37)(30,39,45,47)(35,48,46,42);(1,3,5,14)(2,7,10,15)(4,12,6,13)(8,11,9,16)(17,19)(18,24)(20,35)(21,33)(22,32)(23,31)(25,42)(26,41)(27,29)(28,40)(30,36)(34,47)(37,45)(38,46)(39,43)(44,48);(1,2)(3,11)(4,8)(5,10)(6,9)(7,13)(12,15)(14,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,43)(27,34)(28,44)(29,42)(31,45)(32,35)(33,46)(40,47)(41,48);(1,4,5,6)(2,8,10,9)(3,12,14,13)(7,11,15,16)(19,31)(24,40)(29,41)(30,45)(32,33)(35,46)(39,47)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fhi,g^b=gh,e^c=ei,f^c=fhi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144342);